Existence of eigenvalues embedded in the spectral bands of Schrödinger operators on carbon nanotubes with impurities

  • Hiroaki NiikuniEmail author


It is known in Korotyaev and Lobanov (Ann Henri Poincaré 8:1151–1176, 2007) and Parchment (Commun Math Phys 275:805–826, 2007) that spectra of periodic Schrödinger operators on the metric graph corresponding to a carbon nanotube have the band-gap structure. The band-gap spectrum consists of not only infinitely many closed intervals but also the so-called flat bands (the set of eigenvalue with infinite multiplicities). In this paper, we study the spectrum of the free Schrödinger operators on a zigzag carbon nanotube with finite number of impurities expressed as \(\delta \)-interactions with strength \(\alpha \in \mathbf{R}\). Assume that our impurities are symmetrically located in two ways. For a suitable \(\alpha \), we construct eigenvalues embedded in the interior of spectral bands (not in spectral gaps).


Periodic Schrödinger operators with impurities Carbon nanotube Zigzag nanotube Quantum graph Spectral gap Band structure 

Mathematics Subject Classification

34L05 34L15 34B45 



This work was supported by Grant-in-Aid for Young Scientists (17K14221), Japan Society for Promotion of Science. The author thanks referees for taking their precious time to read the manuscript carefully and giving helpful comments so much to the author. In particular, the author notes that the statements of theorems and lemmas were organized owing to their comments.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Maebashi Institute of TechnologyMaebashi CityJapan

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